LMFDB - Embedded newform 1710.2.d.f.1369.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1710,2,Mod(1369,1710)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1710, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1710.1369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1710.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.6544187456$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 570)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1369.4
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	1710.1369
Dual form			1710.2.d.f.1369.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(-2.21432 - 0.311108i) q^{5} +4.42864i q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +(-2.21432 - 0.311108i) q^{5} +4.42864i q^{7} -1.00000i q^{8} +(0.311108 - 2.21432i) q^{10} -2.62222 q^{11} +5.80642i q^{13} -4.42864 q^{14} +1.00000 q^{16} -3.80642i q^{17} -1.00000 q^{19} +(2.21432 + 0.311108i) q^{20} -2.62222i q^{22} +2.62222i q^{23} +(4.80642 + 1.37778i) q^{25} -5.80642 q^{26} -4.42864i q^{28} +3.37778 q^{29} -4.42864 q^{31} +1.00000i q^{32} +3.80642 q^{34} +(1.37778 - 9.80642i) q^{35} -5.80642i q^{37} -1.00000i q^{38} +(-0.311108 + 2.21432i) q^{40} -5.67307 q^{41} -10.9906i q^{43} +2.62222 q^{44} -2.62222 q^{46} +2.62222i q^{47} -12.6128 q^{49} +(-1.37778 + 4.80642i) q^{50} -5.80642i q^{52} +6.00000i q^{53} +(5.80642 + 0.815792i) q^{55} +4.42864 q^{56} +3.37778i q^{58} -1.05086 q^{59} +4.75557 q^{61} -4.42864i q^{62} -1.00000 q^{64} +(1.80642 - 12.8573i) q^{65} -15.6128i q^{67} +3.80642i q^{68} +(9.80642 + 1.37778i) q^{70} -15.6128 q^{71} +11.6128i q^{73} +5.80642 q^{74} +1.00000 q^{76} -11.6128i q^{77} -4.42864 q^{79} +(-2.21432 - 0.311108i) q^{80} -5.67307i q^{82} -11.9081i q^{83} +(-1.18421 + 8.42864i) q^{85} +10.9906 q^{86} +2.62222i q^{88} +12.4286 q^{89} -25.7146 q^{91} -2.62222i q^{92} -2.62222 q^{94} +(2.21432 + 0.311108i) q^{95} -7.37778i q^{97} -12.6128i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4}+O(q^{10}) $ 6 * q - 6 * q^4	$ 6 q - 6 q^{4} + 2 q^{10} - 16 q^{11} + 6 q^{16} - 6 q^{19} + 2 q^{25} - 8 q^{26} + 20 q^{29} - 4 q^{34} + 8 q^{35} - 2 q^{40} - 8 q^{41} + 16 q^{44} - 16 q^{46} - 22 q^{49} - 8 q^{50} + 8 q^{55} + 20 q^{59} + 28 q^{61} - 6 q^{64} - 16 q^{65} + 32 q^{70} - 40 q^{71} + 8 q^{74} + 6 q^{76} + 20 q^{85} + 12 q^{86} + 48 q^{89} - 48 q^{91} - 16 q^{94}+O(q^{100}) $ 6 * q - 6 * q^4 + 2 * q^10 - 16 * q^11 + 6 * q^16 - 6 * q^19 + 2 * q^25 - 8 * q^26 + 20 * q^29 - 4 * q^34 + 8 * q^35 - 2 * q^40 - 8 * q^41 + 16 * q^44 - 16 * q^46 - 22 * q^49 - 8 * q^50 + 8 * q^55 + 20 * q^59 + 28 * q^61 - 6 * q^64 - 16 * q^65 + 32 * q^70 - 40 * q^71 + 8 * q^74 + 6 * q^76 + 20 * q^85 + 12 * q^86 + 48 * q^89 - 48 * q^91 - 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times$.

$n$	$191$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−2.21432	−	0.311108i	−0.990274	−	0.139132i
$5$	−2.21432	−	0.311108i	−0.990274	−	0.139132i
$6$		0			0
$7$			4.42864i			1.67387i	0.547304	+	0.836934i	$0.315654\pi$
$7$			4.42864i			1.67387i	−0.547304	+	0.836934i	$0.684346\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	0.311108	−	2.21432i	0.0983809	−	0.700229i
$11$	−2.62222			−0.790628			−0.395314	−	0.918546i	$-0.629364\pi$
$11$	−2.62222			−0.790628			−0.395314	+	0.918546i	$0.629364\pi$
$12$		0			0
$13$			5.80642i			1.61041i	0.592995	+	0.805206i	$0.297945\pi$
$13$			5.80642i			1.61041i	−0.592995	+	0.805206i	$0.702055\pi$
$14$	−4.42864			−1.18360
$15$		0			0
$16$	1.00000			0.250000
$17$		−	3.80642i		−	0.923193i	−0.887090	−	0.461597i	$-0.847277\pi$
$17$		−	3.80642i		−	0.923193i	0.887090	−	0.461597i	$-0.152723\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$	2.21432	+	0.311108i	0.495137	+	0.0695658i
$21$		0			0
$22$		−	2.62222i		−	0.559058i
$23$			2.62222i			0.546770i	0.961905	+	0.273385i	$0.0881433\pi$
$23$			2.62222i			0.546770i	−0.961905	+	0.273385i	$0.911857\pi$
$24$		0			0
$25$	4.80642	+	1.37778i	0.961285	+	0.275557i
$26$	−5.80642			−1.13873
$27$		0			0
$28$		−	4.42864i		−	0.836934i
$29$	3.37778			0.627239			0.313619	−	0.949549i	$-0.398458\pi$
$29$	3.37778			0.627239			0.313619	+	0.949549i	$0.398458\pi$
$30$		0			0
$31$	−4.42864			−0.795407			−0.397704	−	0.917514i	$-0.630193\pi$
$31$	−4.42864			−0.795407			−0.397704	+	0.917514i	$0.630193\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	3.80642			0.652796
$35$	1.37778	−	9.80642i	0.232888	−	1.65759i
$36$		0			0
$37$		−	5.80642i		−	0.954570i	−0.878749	−	0.477285i	$-0.841621\pi$
$37$		−	5.80642i		−	0.954570i	0.878749	−	0.477285i	$-0.158379\pi$
$38$		−	1.00000i		−	0.162221i
$39$		0			0
$40$	−0.311108	+	2.21432i	−0.0491905	+	0.350115i
$41$	−5.67307			−0.885985			−0.442992	−	0.896525i	$-0.646083\pi$
$41$	−5.67307			−0.885985			−0.442992	+	0.896525i	$0.646083\pi$
$42$		0			0
$43$		−	10.9906i		−	1.67606i	−0.545627	−	0.838028i	$-0.683709\pi$
$43$		−	10.9906i		−	1.67606i	0.545627	−	0.838028i	$-0.316291\pi$
$44$	2.62222			0.395314
$45$		0			0
$46$	−2.62222			−0.386625
$47$			2.62222i			0.382489i	0.981542	+	0.191245i	$0.0612524\pi$
$47$			2.62222i			0.382489i	−0.981542	+	0.191245i	$0.938748\pi$
$48$		0			0
$49$	−12.6128			−1.80184
$50$	−1.37778	+	4.80642i	−0.194848	+	0.679731i
$51$		0			0
$52$		−	5.80642i		−	0.805206i
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$	5.80642	+	0.815792i	0.782938	+	0.110001i
$56$	4.42864			0.591802
$57$		0			0
$58$			3.37778i			0.443525i
$59$	−1.05086			−0.136810			−0.0684048	−	0.997658i	$-0.521791\pi$
$59$	−1.05086			−0.136810			−0.0684048	+	0.997658i	$0.521791\pi$
$60$		0			0
$61$	4.75557			0.608888			0.304444	−	0.952530i	$-0.401529\pi$
$61$	4.75557			0.608888			0.304444	+	0.952530i	$0.401529\pi$
$62$		−	4.42864i		−	0.562438i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	1.80642	−	12.8573i	0.224059	−	1.59475i
$66$		0			0
$67$		−	15.6128i		−	1.90741i	−0.300739	−	0.953706i	$-0.597233\pi$
$67$		−	15.6128i		−	1.90741i	0.300739	−	0.953706i	$-0.402767\pi$
$68$			3.80642i			0.461597i
$69$		0			0
$70$	9.80642	+	1.37778i	1.17209	+	0.164677i
$71$	−15.6128			−1.85290			−0.926452	−	0.376413i	$-0.877157\pi$
$71$	−15.6128			−1.85290			−0.926452	+	0.376413i	$0.877157\pi$
$72$		0			0
$73$			11.6128i			1.35918i	0.733592	+	0.679591i	$0.237843\pi$
$73$			11.6128i			1.35918i	−0.733592	+	0.679591i	$0.762157\pi$
$74$	5.80642			0.674983
$75$		0			0
$76$	1.00000			0.114708
$77$		−	11.6128i		−	1.32341i
$78$		0			0
$79$	−4.42864			−0.498261			−0.249130	−	0.968470i	$-0.580145\pi$
$79$	−4.42864			−0.498261			−0.249130	+	0.968470i	$0.580145\pi$
$80$	−2.21432	−	0.311108i	−0.247568	−	0.0347829i
$81$		0			0
$82$		−	5.67307i		−	0.626486i
$83$		−	11.9081i		−	1.30709i	−0.756889	−	0.653544i	$-0.773282\pi$
$83$		−	11.9081i		−	1.30709i	0.756889	−	0.653544i	$-0.226718\pi$
$84$		0			0
$85$	−1.18421	+	8.42864i	−0.128445	+	0.914214i
$86$	10.9906			1.18515
$87$		0			0
$88$			2.62222i			0.279529i
$89$	12.4286			1.31743			0.658717	−	0.752391i	$-0.271100\pi$
$89$	12.4286			1.31743			0.658717	+	0.752391i	$0.271100\pi$
$90$		0			0
$91$	−25.7146			−2.69562
$92$		−	2.62222i		−	0.273385i
$93$		0			0
$94$	−2.62222			−0.270461
$95$	2.21432	+	0.311108i	0.227184	+	0.0319190i
$96$		0			0
$97$		−	7.37778i		−	0.749101i	−0.927207	−	0.374550i	$-0.877797\pi$
$97$		−	7.37778i		−	0.749101i	0.927207	−	0.374550i	$-0.122203\pi$
$98$		−	12.6128i		−	1.27409i
$99$		0			0
$100$	−4.80642	−	1.37778i	−0.480642	−	0.137778i
$101$	−17.6731			−1.75854			−0.879268	−	0.476327i	$-0.841968\pi$
$101$	−17.6731			−1.75854			−0.879268	+	0.476327i	$0.841968\pi$
$102$		0			0
$103$		−	1.18421i		−	0.116684i	−0.998297	−	0.0583418i	$-0.981419\pi$
$103$		−	1.18421i		−	0.116684i	0.998297	−	0.0583418i	$-0.0185813\pi$
$104$	5.80642			0.569367
$105$		0			0
$106$	−6.00000			−0.582772
$107$			4.85728i			0.469571i	0.972047	+	0.234785i	$0.0754388\pi$
$107$			4.85728i			0.469571i	−0.972047	+	0.234785i	$0.924561\pi$
$108$		0			0
$109$	6.04149			0.578670			0.289335	−	0.957228i	$-0.406566\pi$
$109$	6.04149			0.578670			0.289335	+	0.957228i	$0.406566\pi$
$110$	−0.815792	+	5.80642i	−0.0777827	+	0.553621i
$111$		0			0
$112$			4.42864i			0.418467i
$113$		−	9.34614i		−	0.879211i	−0.898191	−	0.439606i	$-0.855118\pi$
$113$		−	9.34614i		−	0.879211i	0.898191	−	0.439606i	$-0.144882\pi$
$114$		0			0
$115$	0.815792	−	5.80642i	0.0760730	−	0.541452i
$116$	−3.37778			−0.313619
$117$		0			0
$118$		−	1.05086i		−	0.0967391i
$119$	16.8573			1.54530
$120$		0			0
$121$	−4.12399			−0.374908
$122$			4.75557i			0.430549i
$123$		0			0
$124$	4.42864			0.397704
$125$	−10.2143	−	4.54617i	−0.913597	−	0.406622i
$126$		0			0
$127$		−	6.42864i		−	0.570450i	−0.958461	−	0.285225i	$-0.907932\pi$
$127$		−	6.42864i		−	0.570450i	0.958461	−	0.285225i	$-0.0920682\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	12.8573	+	1.80642i	1.12766	+	0.158434i
$131$	15.0923			1.31862			0.659312	−	0.751869i	$-0.270848\pi$
$131$	15.0923			1.31862			0.659312	+	0.751869i	$0.270848\pi$
$132$		0			0
$133$		−	4.42864i		−	0.384012i
$134$	15.6128			1.34874
$135$		0			0
$136$	−3.80642			−0.326398
$137$		−	7.53972i		−	0.644162i	−0.946712	−	0.322081i	$-0.895618\pi$
$137$		−	7.53972i		−	0.644162i	0.946712	−	0.322081i	$-0.104382\pi$
$138$		0			0
$139$	−0.387152			−0.0328378			−0.0164189	−	0.999865i	$-0.505227\pi$
$139$	−0.387152			−0.0328378			−0.0164189	+	0.999865i	$0.505227\pi$
$140$	−1.37778	+	9.80642i	−0.116444	+	0.828794i
$141$		0			0
$142$		−	15.6128i		−	1.31020i
$143$		−	15.2257i		−	1.27324i
$144$		0			0
$145$	−7.47949	−	1.05086i	−0.621138	−	0.0872688i
$146$	−11.6128			−0.961086
$147$		0			0
$148$			5.80642i			0.477285i
$149$	−14.5303			−1.19037			−0.595186	−	0.803588i	$-0.702922\pi$
$149$	−14.5303			−1.19037			−0.595186	+	0.803588i	$0.702922\pi$
$150$		0			0
$151$	4.69535			0.382102			0.191051	−	0.981580i	$-0.438810\pi$
$151$	4.69535			0.382102			0.191051	+	0.981580i	$0.438810\pi$
$152$			1.00000i			0.0811107i
$153$		0			0
$154$	11.6128			0.935790
$155$	9.80642	+	1.37778i	0.787671	+	0.110666i
$156$		0			0
$157$			21.5210i			1.71756i	0.512343	+	0.858781i	$0.328778\pi$
$157$			21.5210i			1.71756i	−0.512343	+	0.858781i	$0.671222\pi$
$158$		−	4.42864i		−	0.352324i
$159$		0			0
$160$	0.311108	−	2.21432i	0.0245952	−	0.175057i
$161$	−11.6128			−0.915221
$162$		0			0
$163$		−	8.23506i		−	0.645020i	−0.946566	−	0.322510i	$-0.895473\pi$
$163$		−	8.23506i		−	0.645020i	0.946566	−	0.322510i	$-0.104527\pi$
$164$	5.67307			0.442992
$165$		0			0
$166$	11.9081			0.924250
$167$			8.47013i			0.655438i	0.944775	+	0.327719i	$0.106280\pi$
$167$			8.47013i			0.655438i	−0.944775	+	0.327719i	$0.893720\pi$
$168$		0			0
$169$	−20.7146			−1.59343
$170$	−8.42864	−	1.18421i	−0.646447	−	0.0908246i
$171$		0			0
$172$			10.9906i			0.838028i
$173$			22.4701i			1.70837i	0.519967	+	0.854186i	$0.325944\pi$
$173$			22.4701i			1.70837i	−0.519967	+	0.854186i	$0.674056\pi$
$174$		0			0
$175$	−6.10171	+	21.2859i	−0.461246	+	1.60906i
$176$	−2.62222			−0.197657
$177$		0			0
$178$			12.4286i			0.931566i
$179$	−2.94914			−0.220429			−0.110215	−	0.993908i	$-0.535154\pi$
$179$	−2.94914			−0.220429			−0.110215	+	0.993908i	$0.535154\pi$
$180$		0			0
$181$	22.8988			1.70205			0.851026	−	0.525124i	$-0.175981\pi$
$181$	22.8988			1.70205			0.851026	+	0.525124i	$0.175981\pi$
$182$		−	25.7146i		−	1.90609i
$183$		0			0
$184$	2.62222			0.193312
$185$	−1.80642	+	12.8573i	−0.132811	+	0.945286i
$186$		0			0
$187$			9.98126i			0.729902i
$188$		−	2.62222i		−	0.191245i
$189$		0			0
$190$	−0.311108	+	2.21432i	−0.0225701	+	0.160644i
$191$	9.05086			0.654897			0.327448	−	0.944869i	$-0.393811\pi$
$191$	9.05086			0.654897			0.327448	+	0.944869i	$0.393811\pi$
$192$		0			0
$193$			18.7239i			1.34778i	0.738833	+	0.673889i	$0.235377\pi$
$193$			18.7239i			1.34778i	−0.738833	+	0.673889i	$0.764623\pi$
$194$	7.37778			0.529694
$195$		0			0
$196$	12.6128			0.900918
$197$			0.888922i			0.0633331i	0.999498	+	0.0316665i	$0.0100815\pi$
$197$			0.888922i			0.0633331i	−0.999498	+	0.0316665i	$0.989919\pi$
$198$		0			0
$199$	−21.7146			−1.53930			−0.769652	−	0.638464i	$-0.779570\pi$
$199$	−21.7146			−1.53930			−0.769652	+	0.638464i	$0.779570\pi$
$200$	1.37778	−	4.80642i	0.0974241	−	0.339865i
$201$		0			0
$202$		−	17.6731i		−	1.24347i
$203$			14.9590i			1.04992i
$204$		0			0
$205$	12.5620	+	1.76494i	0.877368	+	0.123269i
$206$	1.18421			0.0825077
$207$		0			0
$208$			5.80642i			0.402603i
$209$	2.62222			0.181382
$210$		0			0
$211$	−3.61285			−0.248719			−0.124359	−	0.992237i	$-0.539688\pi$
$211$	−3.61285			−0.248719			−0.124359	+	0.992237i	$0.539688\pi$
$212$		−	6.00000i		−	0.412082i
$213$		0			0
$214$	−4.85728			−0.332037
$215$	−3.41927	+	24.3368i	−0.233192	+	1.65975i
$216$		0			0
$217$		−	19.6128i		−	1.33141i
$218$			6.04149i			0.409181i
$219$		0			0
$220$	−5.80642	−	0.815792i	−0.391469	−	0.0550007i
$221$	22.1017			1.48672
$222$		0			0
$223$		−	21.3876i		−	1.43222i	−0.697987	−	0.716111i	$-0.745921\pi$
$223$		−	21.3876i		−	1.43222i	0.697987	−	0.716111i	$-0.254079\pi$
$224$	−4.42864			−0.295901
$225$		0			0
$226$	9.34614			0.621696
$227$			8.47013i			0.562182i	0.959681	+	0.281091i	$0.0906963\pi$
$227$			8.47013i			0.562182i	−0.959681	+	0.281091i	$0.909304\pi$
$228$		0			0
$229$	16.9590			1.12068			0.560341	−	0.828262i	$-0.310670\pi$
$229$	16.9590			1.12068			0.560341	+	0.828262i	$0.310670\pi$
$230$	5.80642	+	0.815792i	0.382864	+	0.0537917i
$231$		0			0
$232$		−	3.37778i		−	0.221762i
$233$			17.9081i			1.17320i	0.809876	+	0.586600i	$0.199534\pi$
$233$			17.9081i			1.17320i	−0.809876	+	0.586600i	$0.800466\pi$
$234$		0			0
$235$	0.815792	−	5.80642i	0.0532164	−	0.378769i
$236$	1.05086			0.0684048
$237$		0			0
$238$			16.8573i			1.09270i
$239$	−28.2766			−1.82906			−0.914529	−	0.404520i	$-0.867438\pi$
$239$	−28.2766			−1.82906			−0.914529	+	0.404520i	$0.867438\pi$
$240$		0			0
$241$	−11.5111			−0.741498			−0.370749	−	0.928733i	$-0.620899\pi$
$241$	−11.5111			−0.741498			−0.370749	+	0.928733i	$0.620899\pi$
$242$		−	4.12399i		−	0.265100i
$243$		0			0
$244$	−4.75557			−0.304444
$245$	27.9289	+	3.92396i	1.78431	+	0.250692i
$246$		0			0
$247$		−	5.80642i		−	0.369454i
$248$			4.42864i			0.281219i
$249$		0			0
$250$	4.54617	−	10.2143i	0.287525	−	0.646010i
$251$	7.74620			0.488936			0.244468	−	0.969657i	$-0.421387\pi$
$251$	7.74620			0.488936			0.244468	+	0.969657i	$0.421387\pi$
$252$		0			0
$253$		−	6.87601i		−	0.432291i
$254$	6.42864			0.403369
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	13.8796i		−	0.865783i	−0.901446	−	0.432891i	$-0.857493\pi$
$257$		−	13.8796i		−	0.865783i	0.901446	−	0.432891i	$-0.142507\pi$
$258$		0			0
$259$	25.7146			1.59782
$260$	−1.80642	+	12.8573i	−0.112030	+	0.797375i
$261$		0			0
$262$			15.0923i			0.932408i
$263$			18.6222i			1.14830i	0.818752	+	0.574148i	$0.194666\pi$
$263$			18.6222i			1.14830i	−0.818752	+	0.574148i	$0.805334\pi$
$264$		0			0
$265$	1.86665	−	13.2859i	0.114667	−	0.816147i
$266$	4.42864			0.271537
$267$		0			0
$268$			15.6128i			0.953706i
$269$	15.8479			0.966264			0.483132	−	0.875547i	$-0.339499\pi$
$269$	15.8479			0.966264			0.483132	+	0.875547i	$0.339499\pi$
$270$		0			0
$271$	1.51114			0.0917951			0.0458975	−	0.998946i	$-0.485385\pi$
$271$	1.51114			0.0917951			0.0458975	+	0.998946i	$0.485385\pi$
$272$		−	3.80642i		−	0.230798i
$273$		0			0
$274$	7.53972			0.455491
$275$	−12.6035	−	3.61285i	−0.760018	−	0.217863i
$276$		0			0
$277$		−	1.05086i		−	0.0631398i	−0.999502	−	0.0315699i	$-0.989949\pi$
$277$		−	1.05086i		−	0.0631398i	0.999502	−	0.0315699i	$-0.0100507\pi$
$278$		−	0.387152i		−	0.0232199i
$279$		0			0
$280$	−9.80642	−	1.37778i	−0.586046	−	0.0823384i
$281$	−3.45091			−0.205864			−0.102932	−	0.994688i	$-0.532822\pi$
$281$	−3.45091			−0.205864			−0.102932	+	0.994688i	$0.532822\pi$
$282$		0			0
$283$			15.5812i			0.926206i	0.886304	+	0.463103i	$0.153264\pi$
$283$			15.5812i			0.926206i	−0.886304	+	0.463103i	$0.846736\pi$
$284$	15.6128			0.926452
$285$		0			0
$286$	15.2257			0.900314
$287$		−	25.1240i		−	1.48302i
$288$		0			0
$289$	2.51114			0.147714
$290$	1.05086	−	7.47949i	0.0617083	−	0.439211i
$291$		0			0
$292$		−	11.6128i		−	0.679591i
$293$		−	7.12399i		−	0.416188i	−0.978109	−	0.208094i	$-0.933274\pi$
$293$		−	7.12399i		−	0.416188i	0.978109	−	0.208094i	$-0.0667259\pi$
$294$		0			0
$295$	2.32693	+	0.326929i	0.135479	+	0.0190346i
$296$	−5.80642			−0.337492
$297$		0			0
$298$		−	14.5303i		−	0.841721i
$299$	−15.2257			−0.880525
$300$		0			0
$301$	48.6735			2.80550
$302$			4.69535i			0.270187i
$303$		0			0
$304$	−1.00000			−0.0573539
$305$	−10.5303	−	1.47949i	−0.602966	−	0.0847156i
$306$		0			0
$307$		−	1.12399i		−	0.0641492i	−0.999485	−	0.0320746i	$-0.989789\pi$
$307$		−	1.12399i		−	0.0641492i	0.999485	−	0.0320746i	$-0.0102114\pi$
$308$			11.6128i			0.661703i
$309$		0			0
$310$	−1.37778	+	9.80642i	−0.0782529	+	0.556967i
$311$	−14.9491			−0.847688			−0.423844	−	0.905735i	$-0.639320\pi$
$311$	−14.9491			−0.847688			−0.423844	+	0.905735i	$0.639320\pi$
$312$		0			0
$313$			3.14272i			0.177637i	0.996048	+	0.0888185i	$0.0283091\pi$
$313$			3.14272i			0.177637i	−0.996048	+	0.0888185i	$0.971691\pi$
$314$	−21.5210			−1.21450
$315$		0			0
$316$	4.42864			0.249130
$317$			10.5906i			0.594826i	0.954749	+	0.297413i	$0.0961238\pi$
$317$			10.5906i			0.594826i	−0.954749	+	0.297413i	$0.903876\pi$
$318$		0			0
$319$	−8.85728			−0.495912
$320$	2.21432	+	0.311108i	0.123784	+	0.0173915i
$321$		0			0
$322$		−	11.6128i		−	0.647159i
$323$			3.80642i			0.211795i
$324$		0			0
$325$	−8.00000	+	27.9081i	−0.443760	+	1.54806i
$326$	8.23506			0.456098
$327$		0			0
$328$			5.67307i			0.313243i
$329$	−11.6128			−0.640237
$330$		0			0
$331$	−15.1427			−0.832319			−0.416160	−	0.909292i	$-0.636624\pi$
$331$	−15.1427			−0.832319			−0.416160	+	0.909292i	$0.636624\pi$
$332$			11.9081i			0.653544i
$333$		0			0
$334$	−8.47013			−0.463465
$335$	−4.85728	+	34.5718i	−0.265381	+	1.88886i
$336$		0			0
$337$		−	11.1111i		−	0.605259i	−0.953108	−	0.302629i	$-0.902136\pi$
$337$		−	11.1111i		−	0.605259i	0.953108	−	0.302629i	$-0.0978645\pi$
$338$		−	20.7146i		−	1.12672i
$339$		0			0
$340$	1.18421	−	8.42864i	0.0642227	−	0.457107i
$341$	11.6128			0.628871
$342$		0			0
$343$		−	24.8573i		−	1.34217i
$344$	−10.9906			−0.592575
$345$		0			0
$346$	−22.4701			−1.20800
$347$		−	14.1936i		−	0.761951i	−0.924585	−	0.380976i	$-0.875588\pi$
$347$		−	14.1936i		−	0.761951i	0.924585	−	0.380976i	$-0.124412\pi$
$348$		0			0
$349$	−32.3684			−1.73264			−0.866321	−	0.499488i	$-0.833521\pi$
$349$	−32.3684			−1.73264			−0.866321	+	0.499488i	$0.833521\pi$
$350$	−21.2859	−	6.10171i	−1.13778	−	0.326150i
$351$		0			0
$352$		−	2.62222i		−	0.139765i
$353$			11.8064i			0.628393i	0.949358	+	0.314196i	$0.101735\pi$
$353$			11.8064i			0.628393i	−0.949358	+	0.314196i	$0.898265\pi$
$354$		0			0
$355$	34.5718	+	4.85728i	1.83488	+	0.257798i
$356$	−12.4286			−0.658717
$357$		0			0
$358$		−	2.94914i		−	0.155867i
$359$	−10.8287			−0.571517			−0.285758	−	0.958302i	$-0.592245\pi$
$359$	−10.8287			−0.571517			−0.285758	+	0.958302i	$0.592245\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$			22.8988i			1.20353i
$363$		0			0
$364$	25.7146			1.34781
$365$	3.61285	−	25.7146i	0.189105	−	1.34596i
$366$		0			0
$367$			1.46965i			0.0767151i	0.999264	+	0.0383576i	$0.0122126\pi$
$367$			1.46965i			0.0767151i	−0.999264	+	0.0383576i	$0.987787\pi$
$368$			2.62222i			0.136692i
$369$		0			0
$370$	−12.8573	−	1.80642i	−0.668418	−	0.0939115i
$371$	−26.5718			−1.37954
$372$		0			0
$373$			24.3783i			1.26226i	0.775678	+	0.631129i	$0.217408\pi$
$373$			24.3783i			1.26226i	−0.775678	+	0.631129i	$0.782592\pi$
$374$	−9.98126			−0.516119
$375$		0			0
$376$	2.62222			0.135230
$377$			19.6128i			1.01011i
$378$		0			0
$379$	−18.9590			−0.973858			−0.486929	−	0.873442i	$-0.661883\pi$
$379$	−18.9590			−0.973858			−0.486929	+	0.873442i	$0.661883\pi$
$380$	−2.21432	−	0.311108i	−0.113592	−	0.0159595i
$381$		0			0
$382$			9.05086i			0.463082i
$383$			26.1017i			1.33374i	0.745176	+	0.666868i	$0.232365\pi$
$383$			26.1017i			1.33374i	−0.745176	+	0.666868i	$0.767635\pi$
$384$		0			0
$385$	−3.61285	+	25.7146i	−0.184128	+	1.31054i
$386$	−18.7239			−0.953023
$387$		0			0
$388$			7.37778i			0.374550i
$389$	−3.57136			−0.181075			−0.0905376	−	0.995893i	$-0.528859\pi$
$389$	−3.57136			−0.181075			−0.0905376	+	0.995893i	$0.528859\pi$
$390$		0			0
$391$	9.98126			0.504774
$392$			12.6128i			0.637045i
$393$		0			0
$394$	−0.888922			−0.0447832
$395$	9.80642	+	1.37778i	0.493415	+	0.0693239i
$396$		0			0
$397$		−	31.4193i		−	1.57689i	−0.615107	−	0.788444i	$-0.710887\pi$
$397$		−	31.4193i		−	1.57689i	0.615107	−	0.788444i	$-0.289113\pi$
$398$		−	21.7146i		−	1.08845i
$399$		0			0
$400$	4.80642	+	1.37778i	0.240321	+	0.0688892i
$401$	−12.0415			−0.601323			−0.300662	−	0.953731i	$-0.597207\pi$
$401$	−12.0415			−0.601323			−0.300662	+	0.953731i	$0.597207\pi$
$402$		0			0
$403$		−	25.7146i		−	1.28093i
$404$	17.6731			0.879268
$405$		0			0
$406$	−14.9590			−0.742402
$407$			15.2257i			0.754710i
$408$		0			0
$409$	−26.4701			−1.30886			−0.654432	−	0.756121i	$-0.727092\pi$
$409$	−26.4701			−1.30886			−0.654432	+	0.756121i	$0.727092\pi$
$410$	−1.76494	+	12.5620i	−0.0871640	+	0.620393i
$411$		0			0
$412$			1.18421i			0.0583418i
$413$		−	4.65386i		−	0.229001i
$414$		0			0
$415$	−3.70471	+	26.3684i	−0.181857	+	1.29437i
$416$	−5.80642			−0.284683
$417$		0			0
$418$			2.62222i			0.128257i
$419$	−25.9684			−1.26864			−0.634319	−	0.773072i	$-0.718719\pi$
$419$	−25.9684			−1.26864			−0.634319	+	0.773072i	$0.718719\pi$
$420$		0			0
$421$	−29.2672			−1.42640			−0.713198	−	0.700963i	$-0.752754\pi$
$421$	−29.2672			−1.42640			−0.713198	+	0.700963i	$0.752754\pi$
$422$		−	3.61285i		−	0.175871i
$423$		0			0
$424$	6.00000			0.291386
$425$	5.24443	−	18.2953i	0.254392	−	0.887452i
$426$		0			0
$427$			21.0607i			1.01920i
$428$		−	4.85728i		−	0.234785i
$429$		0			0
$430$	−24.3368	−	3.41927i	−1.17362	−	0.164892i
$431$	−14.8385			−0.714747			−0.357374	−	0.933961i	$-0.616328\pi$
$431$	−14.8385			−0.714747			−0.357374	+	0.933961i	$0.616328\pi$
$432$		0			0
$433$		−	5.27607i		−	0.253552i	−0.991931	−	0.126776i	$-0.959537\pi$
$433$		−	5.27607i		−	0.253552i	0.991931	−	0.126776i	$-0.0404629\pi$
$434$	19.6128			0.941447
$435$		0			0
$436$	−6.04149			−0.289335
$437$		−	2.62222i		−	0.125438i
$438$		0			0
$439$	−31.8578			−1.52049			−0.760244	−	0.649638i	$-0.774921\pi$
$439$	−31.8578			−1.52049			−0.760244	+	0.649638i	$0.774921\pi$
$440$	0.815792	−	5.80642i	0.0388913	−	0.276810i
$441$		0			0
$442$			22.1017i			1.05127i
$443$			4.94914i			0.235141i	0.993065	+	0.117570i	$0.0375106\pi$
$443$			4.94914i			0.235141i	−0.993065	+	0.117570i	$0.962489\pi$
$444$		0			0
$445$	−27.5210	−	3.86665i	−1.30462	−	0.183297i
$446$	21.3876			1.01273
$447$		0			0
$448$		−	4.42864i		−	0.209234i
$449$	9.75605			0.460416			0.230208	−	0.973141i	$-0.426059\pi$
$449$	9.75605			0.460416			0.230208	+	0.973141i	$0.426059\pi$
$450$		0			0
$451$	14.8760			0.700484
$452$			9.34614i			0.439606i
$453$		0			0
$454$	−8.47013			−0.397523
$455$	56.9403	+	8.00000i	2.66940	+	0.375046i
$456$		0			0
$457$		−	8.00000i		−	0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$		−	8.00000i		−	0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$			16.9590i			0.792442i
$459$		0			0
$460$	−0.815792	+	5.80642i	−0.0380365	+	0.270726i
$461$	36.1245			1.68248			0.841242	−	0.540659i	$-0.181825\pi$
$461$	36.1245			1.68248			0.841242	+	0.540659i	$0.181825\pi$
$462$		0			0
$463$			35.1209i			1.63221i	0.577905	+	0.816104i	$0.303870\pi$
$463$			35.1209i			1.63221i	−0.577905	+	0.816104i	$0.696130\pi$
$464$	3.37778			0.156810
$465$		0			0
$466$	−17.9081			−0.829578
$467$		−	4.56199i		−	0.211104i	−0.994414	−	0.105552i	$-0.966339\pi$
$467$		−	4.56199i		−	0.211104i	0.994414	−	0.105552i	$-0.0336609\pi$
$468$		0			0
$469$	69.1437			3.19276
$470$	5.80642	+	0.815792i	0.267830	+	0.0376297i
$471$		0			0
$472$			1.05086i			0.0483695i
$473$			28.8198i			1.32514i
$474$		0			0
$475$	−4.80642	−	1.37778i	−0.220534	−	0.0632171i
$476$	−16.8573			−0.772652
$477$		0			0
$478$		−	28.2766i		−	1.29334i
$479$	37.4005			1.70887			0.854437	−	0.519555i	$-0.173902\pi$
$479$	37.4005			1.70887			0.854437	+	0.519555i	$0.173902\pi$
$480$		0			0
$481$	33.7146			1.53725
$482$		−	11.5111i		−	0.524318i
$483$		0			0
$484$	4.12399			0.187454
$485$	−2.29529	+	16.3368i	−0.104224	+	0.741815i
$486$		0			0
$487$			20.9175i			0.947862i	0.880562	+	0.473931i	$0.157165\pi$
$487$			20.9175i			0.947862i	−0.880562	+	0.473931i	$0.842835\pi$
$488$		−	4.75557i		−	0.215274i
$489$		0			0
$490$	−3.92396	+	27.9289i	−0.177266	+	1.26170i
$491$	15.8666			0.716052			0.358026	−	0.933712i	$-0.383450\pi$
$491$	15.8666			0.716052			0.358026	+	0.933712i	$0.383450\pi$
$492$		0			0
$493$		−	12.8573i		−	0.579063i
$494$	5.80642			0.261243
$495$		0			0
$496$	−4.42864			−0.198852
$497$		−	69.1437i		−	3.10152i
$498$		0			0
$499$	−16.7368			−0.749244			−0.374622	−	0.927178i	$-0.622227\pi$
$499$	−16.7368			−0.749244			−0.374622	+	0.927178i	$0.622227\pi$
$500$	10.2143	+	4.54617i	0.456798	+	0.203311i
$501$		0			0
$502$			7.74620i			0.345730i
$503$		−	21.9684i		−	0.979521i	−0.871857	−	0.489760i	$-0.837084\pi$
$503$		−	21.9684i		−	0.979521i	0.871857	−	0.489760i	$-0.162916\pi$
$504$		0			0
$505$	39.1338	+	5.49823i	1.74143	+	0.244668i
$506$	6.87601			0.305676
$507$		0			0
$508$			6.42864i			0.285225i
$509$	−15.2573			−0.676270			−0.338135	−	0.941098i	$-0.609796\pi$
$509$	−15.2573			−0.676270			−0.338135	+	0.941098i	$0.609796\pi$
$510$		0			0
$511$	−51.4291			−2.27509
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	13.8796			0.612201
$515$	−0.368416	+	2.62222i	−0.0162344	+	0.115549i
$516$		0			0
$517$		−	6.87601i		−	0.302407i
$518$			25.7146i			1.12983i
$519$		0			0
$520$	−12.8573	−	1.80642i	−0.563829	−	0.0792169i
$521$	18.3269			0.802917			0.401459	−	0.915877i	$-0.368503\pi$
$521$	18.3269			0.802917			0.401459	+	0.915877i	$0.368503\pi$
$522$		0			0
$523$			24.8573i			1.08693i	0.839431	+	0.543466i	$0.182888\pi$
$523$			24.8573i			1.08693i	−0.839431	+	0.543466i	$0.817112\pi$
$524$	−15.0923			−0.659312
$525$		0			0
$526$	−18.6222			−0.811967
$527$			16.8573i			0.734315i
$528$		0			0
$529$	16.1240			0.701043
$530$	13.2859	+	1.86665i	0.577103	+	0.0810820i
$531$		0			0
$532$			4.42864i			0.192006i
$533$		−	32.9403i		−	1.42680i
$534$		0			0
$535$	1.51114	−	10.7556i	0.0653322	−	0.465004i
$536$	−15.6128			−0.674372
$537$		0			0
$538$			15.8479i			0.683252i
$539$	33.0736			1.42458
$540$		0			0
$541$	−1.34614			−0.0578751			−0.0289376	−	0.999581i	$-0.509212\pi$
$541$	−1.34614			−0.0578751			−0.0289376	+	0.999581i	$0.509212\pi$
$542$			1.51114i			0.0649089i
$543$		0			0
$544$	3.80642			0.163199
$545$	−13.3778	−	1.87955i	−0.573041	−	0.0805112i
$546$		0			0
$547$			8.59057i			0.367306i	0.982991	+	0.183653i	$0.0587923\pi$
$547$			8.59057i			0.367306i	−0.982991	+	0.183653i	$0.941208\pi$
$548$			7.53972i			0.322081i
$549$		0			0
$550$	3.61285	−	12.6035i	0.154052	−	0.537414i
$551$	−3.37778			−0.143898
$552$		0			0
$553$		−	19.6128i		−	0.834023i
$554$	1.05086			0.0446466
$555$		0			0
$556$	0.387152			0.0164189
$557$		−	5.86665i		−	0.248578i	−0.992246	−	0.124289i	$-0.960335\pi$
$557$		−	5.86665i		−	0.248578i	0.992246	−	0.124289i	$-0.0396650\pi$
$558$		0			0
$559$	63.8163			2.69914
$560$	1.37778	−	9.80642i	0.0582220	−	0.414397i
$561$		0			0
$562$		−	3.45091i		−	0.145568i
$563$		−	31.4291i		−	1.32458i	−0.749248	−	0.662290i	$-0.769585\pi$
$563$		−	31.4291i		−	1.32458i	0.749248	−	0.662290i	$-0.230415\pi$
$564$		0			0
$565$	−2.90766	+	20.6953i	−0.122326	+	0.870660i
$566$	−15.5812			−0.654927
$567$		0			0
$568$			15.6128i			0.655101i
$569$	−7.95851			−0.333638			−0.166819	−	0.985988i	$-0.553350\pi$
$569$	−7.95851			−0.333638			−0.166819	+	0.985988i	$0.553350\pi$
$570$		0			0
$571$	−30.8385			−1.29055			−0.645276	−	0.763949i	$-0.723258\pi$
$571$	−30.8385			−1.29055			−0.645276	+	0.763949i	$0.723258\pi$
$572$			15.2257i			0.636618i
$573$		0			0
$574$	25.1240			1.04865
$575$	−3.61285	+	12.6035i	−0.150666	+	0.525601i
$576$		0			0
$577$		−	32.0000i		−	1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$		−	32.0000i		−	1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$			2.51114i			0.104450i
$579$		0			0
$580$	7.47949	+	1.05086i	0.310569	+	0.0436344i
$581$	52.7368			2.18789
$582$		0			0
$583$		−	15.7333i		−	0.651606i
$584$	11.6128			0.480543
$585$		0			0
$586$	7.12399			0.294289
$587$			17.8064i			0.734950i	0.930033	+	0.367475i	$0.119778\pi$
$587$			17.8064i			0.734950i	−0.930033	+	0.367475i	$0.880222\pi$
$588$		0			0
$589$	4.42864			0.182479
$590$	−0.326929	+	2.32693i	−0.0134595	+	0.0957982i
$591$		0			0
$592$		−	5.80642i		−	0.238643i
$593$			13.3176i			0.546887i	0.961888	+	0.273443i	$0.0881626\pi$
$593$			13.3176i			0.546887i	−0.961888	+	0.273443i	$0.911837\pi$
$594$		0			0
$595$	−37.3274	−	5.24443i	−1.53027	−	0.215001i
$596$	14.5303			0.595186
$597$		0			0
$598$		−	15.2257i		−	0.622625i
$599$	−43.8163			−1.79028			−0.895142	−	0.445781i	$-0.852926\pi$
$599$	−43.8163			−1.79028			−0.895142	+	0.445781i	$0.852926\pi$
$600$		0			0
$601$	−9.73329			−0.397029			−0.198515	−	0.980098i	$-0.563612\pi$
$601$	−9.73329			−0.397029			−0.198515	+	0.980098i	$0.563612\pi$
$602$			48.6735i			1.98379i
$603$		0			0
$604$	−4.69535			−0.191051
$605$	9.13182	+	1.28300i	0.371261	+	0.0521615i
$606$		0			0
$607$			31.9398i			1.29640i	0.761472	+	0.648198i	$0.224477\pi$
$607$			31.9398i			1.29640i	−0.761472	+	0.648198i	$0.775523\pi$
$608$		−	1.00000i		−	0.0405554i
$609$		0			0
$610$	1.47949	−	10.5303i	0.0599030	−	0.426361i
$611$	−15.2257			−0.615966
$612$		0			0
$613$			11.8064i			0.476857i	0.971160	+	0.238428i	$0.0766323\pi$
$613$			11.8064i			0.476857i	−0.971160	+	0.238428i	$0.923368\pi$
$614$	1.12399			0.0453603
$615$		0			0
$616$	−11.6128			−0.467895
$617$			26.5620i			1.06935i	0.845059	+	0.534673i	$0.179565\pi$
$617$			26.5620i			1.06935i	−0.845059	+	0.534673i	$0.820435\pi$
$618$		0			0
$619$	−7.34614			−0.295266			−0.147633	−	0.989042i	$-0.547166\pi$
$619$	−7.34614			−0.295266			−0.147633	+	0.989042i	$0.547166\pi$
$620$	−9.80642	−	1.37778i	−0.393835	−	0.0553332i
$621$		0			0
$622$		−	14.9491i		−	0.599406i
$623$			55.0420i			2.20521i
$624$		0			0
$625$	21.2034	+	13.2444i	0.848137	+	0.529777i
$626$	−3.14272			−0.125608
$627$		0			0
$628$		−	21.5210i		−	0.858781i
$629$	−22.1017			−0.881253
$630$		0			0
$631$	−30.7556			−1.22436			−0.612180	−	0.790718i	$-0.709707\pi$
$631$	−30.7556			−1.22436			−0.612180	+	0.790718i	$0.709707\pi$
$632$			4.42864i			0.176162i
$633$		0			0
$634$	−10.5906			−0.420605
$635$	−2.00000	+	14.2351i	−0.0793676	+	0.564901i
$636$		0			0
$637$		−	73.2355i		−	2.90170i
$638$		−	8.85728i		−	0.350663i
$639$		0			0
$640$	−0.311108	+	2.21432i	−0.0122976	+	0.0875287i
$641$	39.3876			1.55572			0.777859	−	0.628439i	$-0.216306\pi$
$641$	39.3876			1.55572			0.777859	+	0.628439i	$0.216306\pi$
$642$		0			0
$643$			24.1146i			0.950988i	0.879719	+	0.475494i	$0.157731\pi$
$643$			24.1146i			0.950988i	−0.879719	+	0.475494i	$0.842269\pi$
$644$	11.6128			0.457610
$645$		0			0
$646$	−3.80642			−0.149762
$647$		−	47.6829i		−	1.87461i	−0.348512	−	0.937304i	$-0.613313\pi$
$647$		−	47.6829i		−	1.87461i	0.348512	−	0.937304i	$-0.386687\pi$
$648$		0			0
$649$	2.75557			0.108166
$650$	−27.9081	−	8.00000i	−1.09465	−	0.313786i
$651$		0			0
$652$			8.23506i			0.322510i
$653$		−	21.7462i		−	0.850995i	−0.904960	−	0.425497i	$-0.860099\pi$
$653$		−	21.7462i		−	0.850995i	0.904960	−	0.425497i	$-0.139901\pi$
$654$		0			0
$655$	−33.4193	−	4.69535i	−1.30580	−	0.183462i
$656$	−5.67307			−0.221496
$657$		0			0
$658$		−	11.6128i		−	0.452716i
$659$	−16.1936			−0.630812			−0.315406	−	0.948957i	$-0.602141\pi$
$659$	−16.1936			−0.630812			−0.315406	+	0.948957i	$0.602141\pi$
$660$		0			0
$661$	−22.5116			−0.875600			−0.437800	−	0.899072i	$-0.644242\pi$
$661$	−22.5116			−0.875600			−0.437800	+	0.899072i	$0.644242\pi$
$662$		−	15.1427i		−	0.588539i
$663$		0			0
$664$	−11.9081			−0.462125
$665$	−1.37778	+	9.80642i	−0.0534282	+	0.380277i
$666$		0			0
$667$			8.85728i			0.342955i
$668$		−	8.47013i		−	0.327719i
$669$		0			0
$670$	−34.5718	−	4.85728i	−1.33563	−	0.187653i
$671$	−12.4701			−0.481404
$672$		0			0
$673$			9.66323i			0.372490i	0.982503	+	0.186245i	$0.0596318\pi$
$673$			9.66323i			0.372490i	−0.982503	+	0.186245i	$0.940368\pi$
$674$	11.1111			0.427983
$675$		0			0
$676$	20.7146			0.796714
$677$		−	6.85728i		−	0.263547i	−0.991280	−	0.131773i	$-0.957933\pi$
$677$		−	6.85728i		−	0.263547i	0.991280	−	0.131773i	$-0.0420671\pi$
$678$		0			0
$679$	32.6735			1.25390
$680$	8.42864	+	1.18421i	0.323224	+	0.0454123i
$681$		0			0
$682$			11.6128i			0.444679i
$683$		−	30.3051i		−	1.15959i	−0.814761	−	0.579797i	$-0.803132\pi$
$683$		−	30.3051i		−	1.15959i	0.814761	−	0.579797i	$-0.196868\pi$
$684$		0			0
$685$	−2.34567	+	16.6953i	−0.0896233	+	0.637896i
$686$	24.8573			0.949055
$687$		0			0
$688$		−	10.9906i		−	0.419014i
$689$	−34.8385			−1.32724
$690$		0			0
$691$	7.61285			0.289606			0.144803	−	0.989460i	$-0.453745\pi$
$691$	7.61285			0.289606			0.144803	+	0.989460i	$0.453745\pi$
$692$		−	22.4701i		−	0.854186i
$693$		0			0
$694$	14.1936			0.538781
$695$	0.857279	+	0.120446i	0.0325184	+	0.00456878i
$696$		0			0
$697$			21.5941i			0.817935i
$698$		−	32.3684i		−	1.22516i
$699$		0			0
$700$	6.10171	−	21.2859i	0.230623	−	0.804532i
$701$	32.3654			1.22242			0.611211	−	0.791467i	$-0.290683\pi$
$701$	32.3654			1.22242			0.611211	+	0.791467i	$0.290683\pi$
$702$		0			0
$703$			5.80642i			0.218993i
$704$	2.62222			0.0988285
$705$		0			0
$706$	−11.8064			−0.444341
$707$		−	78.2677i		−	2.94356i
$708$		0			0
$709$	49.8992			1.87401			0.937003	−	0.349322i	$-0.113588\pi$
$709$	49.8992			1.87401			0.937003	+	0.349322i	$0.113588\pi$
$710$	−4.85728	+	34.5718i	−0.182290	+	1.29746i
$711$		0			0
$712$		−	12.4286i		−	0.465783i
$713$		−	11.6128i		−	0.434905i
$714$		0			0
$715$	−4.73683	+	33.7146i	−0.177148	+	1.26085i
$716$	2.94914			0.110215
$717$		0			0
$718$		−	10.8287i		−	0.404123i
$719$	−26.4415			−0.986103			−0.493052	−	0.870000i	$-0.664119\pi$
$719$	−26.4415			−0.986103			−0.493052	+	0.870000i	$0.664119\pi$
$720$		0			0
$721$	5.24443			0.195313
$722$			1.00000i			0.0372161i
$723$		0			0
$724$	−22.8988			−0.851026
$725$	16.2351	+	4.65386i	0.602955	+	0.172840i
$726$		0			0
$727$		−	48.1245i		−	1.78484i	−0.451208	−	0.892419i	$-0.649007\pi$
$727$		−	48.1245i		−	1.78484i	0.451208	−	0.892419i	$-0.350993\pi$
$728$			25.7146i			0.953045i
$729$		0			0
$730$	25.7146	+	3.61285i	0.951738	+	0.133717i
$731$	−41.8350			−1.54732
$732$		0			0
$733$		−	14.0286i		−	0.518157i	−0.965856	−	0.259079i	$-0.916581\pi$
$733$		−	14.0286i		−	0.518157i	0.965856	−	0.259079i	$-0.0834189\pi$
$734$	−1.46965			−0.0542458
$735$		0			0
$736$	−2.62222			−0.0966562
$737$			40.9403i			1.50805i
$738$		0			0
$739$	−10.1847			−0.374650			−0.187325	−	0.982298i	$-0.559982\pi$
$739$	−10.1847			−0.374650			−0.187325	+	0.982298i	$0.559982\pi$
$740$	1.80642	−	12.8573i	0.0664055	−	0.472643i
$741$		0			0
$742$		−	26.5718i		−	0.975483i
$743$			42.9590i			1.57601i	0.615667	+	0.788006i	$0.288887\pi$
$743$			42.9590i			1.57601i	−0.615667	+	0.788006i	$0.711113\pi$
$744$		0			0
$745$	32.1748	+	4.52051i	1.17879	+	0.165619i
$746$	−24.3783			−0.892552
$747$		0			0
$748$		−	9.98126i		−	0.364951i
$749$	−21.5111			−0.786000
$750$		0			0
$751$	−7.18421			−0.262155			−0.131078	−	0.991372i	$-0.541844\pi$
$751$	−7.18421			−0.262155			−0.131078	+	0.991372i	$0.541844\pi$
$752$			2.62222i			0.0956224i
$753$		0			0
$754$	−19.6128			−0.714258
$755$	−10.3970	−	1.46076i	−0.378385	−	0.0531625i
$756$		0			0
$757$		−	41.2543i		−	1.49941i	−0.661771	−	0.749706i	$-0.730195\pi$
$757$		−	41.2543i		−	1.49941i	0.661771	−	0.749706i	$-0.269805\pi$
$758$		−	18.9590i		−	0.688621i
$759$		0			0
$760$	0.311108	−	2.21432i	0.0112851	−	0.0803218i
$761$	−44.3051			−1.60606			−0.803030	−	0.595939i	$-0.796780\pi$
$761$	−44.3051			−1.60606			−0.803030	+	0.595939i	$0.796780\pi$
$762$		0			0
$763$			26.7556i			0.968617i
$764$	−9.05086			−0.327448
$765$		0			0
$766$	−26.1017			−0.943093
$767$		−	6.10171i		−	0.220320i
$768$		0			0
$769$	6.59057			0.237662			0.118831	−	0.992914i	$-0.462085\pi$
$769$	6.59057			0.237662			0.118831	+	0.992914i	$0.462085\pi$
$770$	−25.7146	−	3.61285i	−0.926688	−	0.130198i
$771$		0			0
$772$		−	18.7239i		−	0.673889i
$773$			8.83854i			0.317900i	0.987287	+	0.158950i	$0.0508109\pi$
$773$			8.83854i			0.317900i	−0.987287	+	0.158950i	$0.949189\pi$
$774$		0			0
$775$	−21.2859	−	6.10171i	−0.764613	−	0.219180i
$776$	−7.37778			−0.264847
$777$		0			0
$778$		−	3.57136i		−	0.128039i
$779$	5.67307			0.203259
$780$		0			0
$781$	40.9403			1.46496
$782$			9.98126i			0.356929i
$783$		0			0
$784$	−12.6128			−0.450459
$785$	6.69535	−	47.6543i	0.238967	−	1.70086i
$786$		0			0
$787$			16.9403i			0.603855i	0.953331	+	0.301927i	$0.0976300\pi$
$787$			16.9403i			0.603855i	−0.953331	+	0.301927i	$0.902370\pi$
$788$		−	0.888922i		−	0.0316665i
$789$		0			0
$790$	−1.37778	+	9.80642i	−0.0490194	+	0.348897i
$791$	41.3907			1.47168
$792$		0			0
$793$			27.6128i			0.980561i
$794$	31.4193			1.11503
$795$		0			0
$796$	21.7146			0.769652
$797$		−	40.1847i		−	1.42341i	−0.702476	−	0.711707i	$-0.747922\pi$
$797$		−	40.1847i		−	1.42341i	0.702476	−	0.711707i	$-0.252078\pi$
$798$		0			0
$799$	9.98126			0.353112
$800$	−1.37778	+	4.80642i	−0.0487120	+	0.169933i
$801$		0			0
$802$		−	12.0415i		−	0.425200i
$803$		−	30.4514i		−	1.07461i
$804$		0			0
$805$	25.7146	+	3.61285i	0.906319	+	0.127336i
$806$	25.7146			0.905757
$807$		0			0
$808$			17.6731i			0.621736i
$809$	29.1052			1.02329			0.511643	−	0.859198i	$-0.329037\pi$
$809$	29.1052			1.02329			0.511643	+	0.859198i	$0.329037\pi$
$810$		0			0
$811$	38.7753			1.36158			0.680792	−	0.732477i	$-0.261636\pi$
$811$	38.7753			1.36158			0.680792	+	0.732477i	$0.261636\pi$
$812$		−	14.9590i		−	0.524958i
$813$		0			0
$814$	−15.2257			−0.533660
$815$	−2.56199	+	18.2351i	−0.0897427	+	0.638746i
$816$		0			0
$817$			10.9906i			0.384514i
$818$		−	26.4701i		−	0.925506i
$819$		0			0
$820$	−12.5620	−	1.76494i	−0.438684	−	0.0616343i
$821$	−2.53035			−0.0883098			−0.0441549	−	0.999025i	$-0.514060\pi$
$821$	−2.53035			−0.0883098			−0.0441549	+	0.999025i	$0.514060\pi$
$822$		0			0
$823$		−	24.0415i		−	0.838034i	−0.907978	−	0.419017i	$-0.862375\pi$
$823$		−	24.0415i		−	0.838034i	0.907978	−	0.419017i	$-0.137625\pi$
$824$	−1.18421			−0.0412538
$825$		0			0
$826$	4.65386			0.161928
$827$		−	6.57184i		−	0.228525i	−0.993451	−	0.114263i	$-0.963549\pi$
$827$		−	6.57184i		−	0.228525i	0.993451	−	0.114263i	$-0.0364505\pi$
$828$		0			0
$829$	28.7338			0.997965			0.498983	−	0.866612i	$-0.333707\pi$
$829$	28.7338			0.997965			0.498983	+	0.866612i	$0.333707\pi$
$830$	−26.3684	−	3.70471i	−0.915261	−	0.128592i
$831$		0			0
$832$		−	5.80642i		−	0.201302i
$833$			48.0098i			1.66344i
$834$		0			0
$835$	2.63512	−	18.7556i	0.0911922	−	0.649063i
$836$	−2.62222			−0.0906912
$837$		0			0
$838$		−	25.9684i		−	0.897062i
$839$	−50.5718			−1.74593			−0.872967	−	0.487780i	$-0.837807\pi$
$839$	−50.5718			−1.74593			−0.872967	+	0.487780i	$0.837807\pi$
$840$		0			0
$841$	−17.5906			−0.606571
$842$		−	29.2672i		−	1.00861i
$843$		0			0
$844$	3.61285			0.124359
$845$	45.8687	+	6.44446i	1.57793	+	0.221696i
$846$		0			0
$847$		−	18.2636i		−	0.627546i
$848$			6.00000i			0.206041i
$849$		0			0
$850$	18.2953	+	5.24443i	0.627523	+	0.179883i
$851$	15.2257			0.521930
$852$		0			0
$853$		−	31.2355i		−	1.06948i	−0.845015	−	0.534742i	$-0.820409\pi$
$853$		−	31.2355i		−	1.06948i	0.845015	−	0.534742i	$-0.179591\pi$
$854$	−21.0607			−0.720682
$855$		0			0
$856$	4.85728			0.166018
$857$			27.3274i			0.933486i	0.884393	+	0.466743i	$0.154573\pi$
$857$			27.3274i			0.933486i	−0.884393	+	0.466743i	$0.845427\pi$
$858$		0			0
$859$	34.7753			1.18652			0.593258	−	0.805012i	$-0.297841\pi$
$859$	34.7753			1.18652			0.593258	+	0.805012i	$0.297841\pi$
$860$	3.41927	−	24.3368i	0.116596	−	0.829877i
$861$		0			0
$862$		−	14.8385i		−	0.505403i
$863$		−	21.2444i		−	0.723169i	−0.932339	−	0.361584i	$-0.882236\pi$
$863$		−	21.2444i		−	0.723169i	0.932339	−	0.361584i	$-0.117764\pi$
$864$		0			0
$865$	6.99063	−	49.7560i	0.237689	−	1.69176i
$866$	5.27607			0.179288
$867$		0			0
$868$			19.6128i			0.665703i
$869$	11.6128			0.393939
$870$		0			0
$871$	90.6548			3.07172
$872$		−	6.04149i		−	0.204591i
$873$		0			0
$874$	2.62222			0.0886978
$875$	20.1334	−	45.2355i	0.680632	−	1.52924i
$876$		0			0
$877$		−	22.3970i		−	0.756293i	−0.925746	−	0.378146i	$-0.876562\pi$
$877$		−	22.3970i		−	0.756293i	0.925746	−	0.378146i	$-0.123438\pi$
$878$		−	31.8578i		−	1.07515i
$879$		0			0
$880$	5.80642	+	0.815792i	0.195735	+	0.0275003i
$881$	20.3684			0.686229			0.343115	−	0.939294i	$-0.388518\pi$
$881$	20.3684			0.686229			0.343115	+	0.939294i	$0.388518\pi$
$882$		0			0
$883$			21.6829i			0.729688i	0.931069	+	0.364844i	$0.118878\pi$
$883$			21.6829i			0.729688i	−0.931069	+	0.364844i	$0.881122\pi$
$884$	−22.1017			−0.743361
$885$		0			0
$886$	−4.94914			−0.166270
$887$		−	7.14272i		−	0.239829i	−0.992784	−	0.119915i	$-0.961738\pi$
$887$		−	7.14272i		−	0.239829i	0.992784	−	0.119915i	$-0.0382621\pi$
$888$		0			0
$889$	28.4701			0.954857
$890$	3.86665	−	27.5210i	0.129610	−	0.922505i
$891$		0			0
$892$			21.3876i			0.716111i
$893$		−	2.62222i		−	0.0877491i
$894$		0			0
$895$	6.53035	+	0.917502i	0.218286	+	0.0306687i
$896$	4.42864			0.147950
$897$		0			0
$898$			9.75605i			0.325563i
$899$	−14.9590			−0.498910
$900$		0			0
$901$	22.8385			0.760862
$902$			14.8760i			0.495317i
$903$		0			0
$904$	−9.34614			−0.310848
$905$	−50.7052	−	7.12399i	−1.68550	−	0.236809i
$906$		0			0
$907$		−	15.3461i		−	0.509560i	−0.966999	−	0.254780i	$-0.917997\pi$
$907$		−	15.3461i		−	0.509560i	0.966999	−	0.254780i	$-0.0820031\pi$
$908$		−	8.47013i		−	0.281091i
$909$		0			0
$910$	−8.00000	+	56.9403i	−0.265197	+	1.88755i
$911$	−12.1204			−0.401568			−0.200784	−	0.979636i	$-0.564349\pi$
$911$	−12.1204			−0.401568			−0.200784	+	0.979636i	$0.564349\pi$
$912$		0			0
$913$			31.2257i			1.03342i
$914$	8.00000			0.264616
$915$		0			0
$916$	−16.9590			−0.560341
$917$			66.8385i			2.20720i
$918$		0			0
$919$	12.2667			0.404641			0.202321	−	0.979319i	$-0.435152\pi$
$919$	12.2667			0.404641			0.202321	+	0.979319i	$0.435152\pi$
$920$	−5.80642	−	0.815792i	−0.191432	−	0.0268959i
$921$		0			0
$922$			36.1245i			1.18970i
$923$		−	90.6548i		−	2.98394i
$924$		0			0
$925$	8.00000	−	27.9081i	0.263038	−	0.917614i
$926$	−35.1209			−1.15415
$927$		0			0
$928$			3.37778i			0.110881i
$929$	−40.1847			−1.31842			−0.659208	−	0.751960i	$-0.729108\pi$
$929$	−40.1847			−1.31842			−0.659208	+	0.751960i	$0.729108\pi$
$930$		0			0
$931$	12.6128			0.413369
$932$		−	17.9081i		−	0.586600i
$933$		0			0
$934$	4.56199			0.149273
$935$	3.10525	−	22.1017i	0.101553	−	0.722803i
$936$		0			0
$937$			24.7368i			0.808117i	0.914733	+	0.404059i	$0.132401\pi$
$937$			24.7368i			0.808117i	−0.914733	+	0.404059i	$0.867599\pi$
$938$			69.1437i			2.25762i
$939$		0			0
$940$	−0.815792	+	5.80642i	−0.0266082	+	0.189385i
$941$	1.66323			0.0542196			0.0271098	−	0.999632i	$-0.491370\pi$
$941$	1.66323			0.0542196			0.0271098	+	0.999632i	$0.491370\pi$
$942$		0			0
$943$		−	14.8760i		−	0.484430i
$944$	−1.05086			−0.0342024
$945$		0			0
$946$	−28.8198			−0.937013
$947$			14.8671i			0.483117i	0.970386	+	0.241558i	$0.0776585\pi$
$947$			14.8671i			0.483117i	−0.970386	+	0.241558i	$0.922341\pi$
$948$		0			0
$949$	−67.4291			−2.18884
$950$	1.37778	−	4.80642i	0.0447012	−	0.155941i
$951$		0			0
$952$		−	16.8573i		−	0.546348i
$953$			8.83854i			0.286308i	0.989700	+	0.143154i	$0.0457245\pi$
$953$			8.83854i			0.286308i	−0.989700	+	0.143154i	$0.954275\pi$
$954$		0			0
$955$	−20.0415	−	2.81579i	−0.648527	−	0.0911169i
$956$	28.2766			0.914529
$957$		0			0
$958$			37.4005i			1.20836i
$959$	33.3907			1.07824
$960$		0			0
$961$	−11.3872			−0.367327
$962$			33.7146i			1.08700i
$963$		0			0
$964$	11.5111			0.370749
$965$	5.82516	−	41.4608i	0.187519	−	1.33467i
$966$		0			0
$967$		−	1.55262i		−	0.0499290i	−0.999688	−	0.0249645i	$-0.992053\pi$
$967$		−	1.55262i		−	0.0499290i	0.999688	−	0.0249645i	$-0.00794728\pi$
$968$			4.12399i			0.132550i
$969$		0			0
$970$	−16.3368	−	2.29529i	−0.524542	−	0.0736972i
$971$	−44.5433			−1.42946			−0.714731	−	0.699400i	$-0.753451\pi$
$971$	−44.5433			−1.42946			−0.714731	+	0.699400i	$0.753451\pi$
$972$		0			0
$973$		−	1.71456i		−	0.0549662i
$974$	−20.9175			−0.670240
$975$		0			0
$976$	4.75557			0.152222
$977$			15.8350i			0.506607i	0.967387	+	0.253303i	$0.0815171\pi$
$977$			15.8350i			0.506607i	−0.967387	+	0.253303i	$0.918483\pi$
$978$		0			0
$979$	−32.5906			−1.04160
$980$	−27.9289	−	3.92396i	−0.892155	−	0.125346i
$981$		0			0
$982$			15.8666i			0.506325i
$983$		−	6.87601i		−	0.219311i	−0.993970	−	0.109655i	$-0.965025\pi$
$983$		−	6.87601i		−	0.219311i	0.993970	−	0.109655i	$-0.0349747\pi$
$984$		0			0
$985$	0.276551	−	1.96836i	0.00881163	−	0.0627171i
$986$	12.8573			0.409459
$987$		0			0
$988$			5.80642i			0.184727i
$989$	28.8198			0.916417
$990$		0			0
$991$	−29.8765			−0.949058			−0.474529	−	0.880240i	$-0.657382\pi$
$991$	−29.8765			−0.949058			−0.474529	+	0.880240i	$0.657382\pi$
$992$		−	4.42864i		−	0.140609i
$993$		0			0
$994$	69.1437			2.19310
$995$	48.0830	+	6.75557i	1.52433	+	0.214166i
$996$		0			0
$997$		−	5.90813i		−	0.187112i	−0.995614	−	0.0935562i	$-0.970177\pi$
$997$		−	5.90813i		−	0.187112i	0.995614	−	0.0935562i	$-0.0298235\pi$
$998$		−	16.7368i		−	0.529795i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1710.2.d.f.1369.4		6
3.2	odd	2		570.2.d.c.229.3	✓	6
5.2	odd	4		8550.2.a.ce.1.1		3
5.3	odd	4		8550.2.a.cq.1.3		3
5.4	even	2	inner	1710.2.d.f.1369.1		6
15.2	even	4		2850.2.a.bm.1.1		3
15.8	even	4		2850.2.a.bl.1.3		3
15.14	odd	2		570.2.d.c.229.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.d.c.229.3	✓	6	3.2	odd	2
570.2.d.c.229.6	yes	6	15.14	odd	2
1710.2.d.f.1369.1		6	5.4	even	2	inner
1710.2.d.f.1369.4		6	1.1	even	1	trivial
2850.2.a.bl.1.3		3	15.8	even	4
2850.2.a.bm.1.1		3	15.2	even	4
8550.2.a.ce.1.1		3	5.2	odd	4
8550.2.a.cq.1.3		3	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1710.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-2.21432 - 0.311108i) q^{5} +4.42864i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-2.21432 - 0.311108i) q^{5} +4.42864i q^{7} -1.00000i q^{8} +(0.311108 - 2.21432i) q^{10} -2.62222 q^{11} +5.80642i q^{13} -4.42864 q^{14} +1.00000 q^{16} -3.80642i q^{17} -1.00000 q^{19} +(2.21432 + 0.311108i) q^{20} -2.62222i q^{22} +2.62222i q^{23} +(4.80642 + 1.37778i) q^{25} -5.80642 q^{26} -4.42864i q^{28} +3.37778 q^{29} -4.42864 q^{31} +1.00000i q^{32} +3.80642 q^{34} +(1.37778 - 9.80642i) q^{35} -5.80642i q^{37} -1.00000i q^{38} +(-0.311108 + 2.21432i) q^{40} -5.67307 q^{41} -10.9906i q^{43} +2.62222 q^{44} -2.62222 q^{46} +2.62222i q^{47} -12.6128 q^{49} +(-1.37778 + 4.80642i) q^{50} -5.80642i q^{52} +6.00000i q^{53} +(5.80642 + 0.815792i) q^{55} +4.42864 q^{56} +3.37778i q^{58} -1.05086 q^{59} +4.75557 q^{61} -4.42864i q^{62} -1.00000 q^{64} +(1.80642 - 12.8573i) q^{65} -15.6128i q^{67} +3.80642i q^{68} +(9.80642 + 1.37778i) q^{70} -15.6128 q^{71} +11.6128i q^{73} +5.80642 q^{74} +1.00000 q^{76} -11.6128i q^{77} -4.42864 q^{79} +(-2.21432 - 0.311108i) q^{80} -5.67307i q^{82} -11.9081i q^{83} +(-1.18421 + 8.42864i) q^{85} +10.9906 q^{86} +2.62222i q^{88} +12.4286 q^{89} -25.7146 q^{91} -2.62222i q^{92} -2.62222 q^{94} +(2.21432 + 0.311108i) q^{95} -7.37778i q^{97} -12.6128i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4}+O(q^{10}) \) 6 * q - 6 * q^4	\( 6 q - 6 q^{4} + 2 q^{10} - 16 q^{11} + 6 q^{16} - 6 q^{19} + 2 q^{25} - 8 q^{26} + 20 q^{29} - 4 q^{34} + 8 q^{35} - 2 q^{40} - 8 q^{41} + 16 q^{44} - 16 q^{46} - 22 q^{49} - 8 q^{50} + 8 q^{55} + 20 q^{59} + 28 q^{61} - 6 q^{64} - 16 q^{65} + 32 q^{70} - 40 q^{71} + 8 q^{74} + 6 q^{76} + 20 q^{85} + 12 q^{86} + 48 q^{89} - 48 q^{91} - 16 q^{94}+O(q^{100}) \) 6 * q - 6 * q^4 + 2 * q^10 - 16 * q^11 + 6 * q^16 - 6 * q^19 + 2 * q^25 - 8 * q^26 + 20 * q^29 - 4 * q^34 + 8 * q^35 - 2 * q^40 - 8 * q^41 + 16 * q^44 - 16 * q^46 - 22 * q^49 - 8 * q^50 + 8 * q^55 + 20 * q^59 + 28 * q^61 - 6 * q^64 - 16 * q^65 + 32 * q^70 - 40 * q^71 + 8 * q^74 + 6 * q^76 + 20 * q^85 + 12 * q^86 + 48 * q^89 - 48 * q^91 - 16 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more